Modified kernel for image interpolation

ABSTRACT

A method of providing a representation of image data is disclosed. The method accesses a plurality of discrete sample values of the image data and calculates kernel values for each of the discrete sample values using a scaled kernel. The scaled kernel is constructed by transforming a kernel from a first range to a second range. In order to provide a representation of the image data, the kernel values are convolved with the discrete sample values.

FIELD OF INVENTION

The present invention relates to a method of resolution conversion ofdigital data and in particular digital image and video data. Theinvention also relates to a computer program product including acomputer readable medium having recorded thereon a computer program forresolution conversion of digital data.

BACKGROUND OF INVENTION

A filter function used in digital data resolution conversion is oftencalled a convolution kernel, a filter kernel or just a kernel. When akernel produces data that passes through original data points of asampled signal, it is often called an interpolating kernel and when theinterpolated data produced is not constrained to pass through theoriginal data points it is often called an approximating kernel.

Prior art kernels for digital data resolution conversion include thenearest neighbour (NN), linear, quadratic and cubic kernels. Coefficientvalue versus pixel plots of these kernels are shown in FIGS. 1 to 4respectively. The NN kernel is the simplest method of interpolation,simply interpolating the image with the pixel value that is spatiallynearest to the required pixel value. This method works quite well whenthe scaling ratio is an integer multiple of the original data as itintroduces no new values, ie. no new colours, and preserves sharp edges.However, at other ratios the NN kernel has the disadvantage of shiftingedge locations which often produces visible distortions in the outputimage, especially in images containing text or fine line details. Linearinterpolation on the other hand allows for the introduction of new greylevels (or colours) that are effectively used to position edges atsub-pixel locations. This has the advantage of reducing the effect ofshifted edge locations, however sharp edges can now appear to beblurred. Quadratic and cubic interpolation provide steeper stepresponses and therefore less edge blurring, however, the steeperresponse results in an overshoot on either side of the edge. Theseovershoots can make the edges in natural images appear sharper, but ontext, fine lines, or on other computer generated graphics theseovershoots are clearly visible and detract from the perceived imagequality and text legibility. Step responses of these four kernels areshown in FIGS. 5 to 8.

A digital data resolution conversion method is known in which theparameters of a cubic kernel are adjusted so as to remove the overshootin a step response using a two parameter Catmul-Rom cubic that has akernel h(s) of the form: $\begin{matrix}{{h(s)} = \left\{ \quad \begin{matrix}{{{\left( {2 - {\frac{3}{2}\quad b} - c} \right)\quad {s}^{3}} + {\left( {{- 3} + {2b} + c} \right)\quad {s}^{2}} + \left( {1 - {\frac{1}{3}b}} \right)},{{s} \leq 1}} \\{{{\left( {{\frac{1}{6}\quad b} - c} \right)\quad {s}^{3}} + {\left( {b + {5c}} \right)\quad {s}^{2}} + {\left( {{{- 2}b} - {8c}} \right)\quad {s}} + \left( {{\frac{4}{3}b} + {4c}} \right)},{1 < {s} \leq 2}} \\{0,\quad {Otherwise}}\end{matrix}\quad \right.} & \text{(1)}\end{matrix}$

where s=t/Δt, the parameter b is fixed at b=0, whilst c is variedbetween 0, 0.5, and 1 dependent upon the edge strength (ie: thesharpness of the edge) measured using a Laplacian of Gaussian (LOG) edgedetector. At a sharp edge c=0 and the resulting cubic is:$\begin{matrix}{{h(s)} = \left\{ \quad \begin{matrix}{{{2{s}^{3}} - {3{s}^{2}} + 1},{{s} \leq 1}} \\{0,\quad {Otherwise}}\end{matrix}\quad \right.} & \text{(2)}\end{matrix}$

There is however, a problem with using this kernel to interpolate imagedata when the resampled pixel displacement is not significantlydifferent from the original pixel displacement, say a re-sampling ratioof 10/11 or 11/10. In this case, pixels at the edges of text and otherfine lines take on a grey value rather than the original black or whiteimage values. This again results in a perceived blurring of sharp edgesand a reduction in the observed image quality.

It is an object of the present invention to ameliorate one or moredisadvantages of the prior art.

SUMMARY OF THE INVENTION

According to one aspect of the invention there is provided a method ofproviding a representation of image data, the method comprising thefollowing steps:

(i) accessing a plurality of discrete sample values of said image data;

(ii) calculating kernel values for each of said discrete sample valuesusing a scaled kernel, wherein said scaled kernel is constructed bytransforming a kernel from a first range to a second range, and whereinsaid second range is less than said first range, wherein kernel valuescan be calculated over the said entire first range; and

(iii) convolving said kernel values with said discrete sample values toprovide a representation of said image data.

According to another aspect of the invention there is provided anapparatus for providing a representation of image data, the apparatuscomprising:

accessing means for accessing a plurality of discrete sample values ofsaid image data;

calculating means for calculating kernel values for each of saiddiscrete sample values using a scaled kernel, wherein said scaled kernelis constructed by transforming a kernel from a first range to a secondrange, and wherein said second range is less than said first range,wherein kernel values can be calculated over the said entire firstrange; and

convolving means for convolving said kernel values with said discretesample values to provide a representation of said image data.

According to a still further aspect of the present invention there isprovided a computer readable medium for storing a program for anapparatus which processes data, said processing comprising a method ofproviding a representation of image data, said program comprising:

code for accessing a plurality of discrete sample values of said imagedata;

code for calculating kernel values for each of said discrete samplevalues using a scaled kernel, wherein said scaled kernel is constructedby transforming a kernel from a first range to a second range, andwherein said second range is less than said first range, wherein kernelvalues can be calculated over the said entire first range; and

code for convolving said kernel values with said discrete sample valuesto provide a representation of said image data.

According to a still further aspect of the present invention there isprovided a method of representing image data, the method comprising thefollowing steps:

(i) accessing a plurality of discrete sample values of said image data;

(ii) calculating kernel values for each of said discrete sample valuesusing a scaled kernel, wherein said scaled kernel is of the form:${h(s)} = \left\{ \begin{matrix}{{1 - d} < s \leq d} & \quad \\{0,{{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)}} & \quad \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},\quad {otherwise}} & \quad\end{matrix} \right.$

 and wherein s=t/Δt and 0≦d<0.5; and

(iii) convolving said kernel values with said discrete sample values toprovide a representation of said image data.

According to a still further aspect of the present invention there isprovided an apparatus for representing image data, the apparatuscomprising:

accessing means for accessing a plurality of discrete sample values ofsaid image data;

calculating means for calculating kernel values for each of saiddiscrete sample values using a scaled kernel, wherein said scaled kernelis of the form: ${h(s)} = \left\{ \begin{matrix}{{1 - d} < s \leq d} & \quad \\{0,{{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)}} & \quad \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},\quad {otherwise}} & \quad\end{matrix} \right.$

 and wherein s=t/Δt and 0≦d<0.5; and

convolving means for convolving said kernel values with said discretesample values to provide a representation of said image data.

According to a still further aspect of the present invention there isprovided a computer readable medium for storing a program for anapparatus which processes data, said processing comprising a method ofrepresenting image data, said program comprising:

code for accessing a plurality of discrete sample values of said imagedata;

code for calculating kernel values for each of said discrete samplevalues using a scaled kernel, wherein said scaled kernel is of the form:${h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.$

 otherwise and wherein s=t/Δt and 0≦d<0.5; and

code for convolving said kernel values with said discrete sample valuesto provide a representation of said image data.

According to a still further aspect of the present invention there isprovided a method of transforming a kernel for image processing themethod comprising the following steps:

(i) accessing a kernel for image processing;

(ii) transforming said kernel from a first range to a second range,wherein said second range is less than said first range; and

(iii) storing said transformed kernel.

According to a still further aspect of the present invention there isprovided an apparatus for transforming a kernel for image processing,said apparatus comprising:

accessing means for accessing a kernel for image processing;

transforming means for transforming said kernel from a first range to asecond range, wherein said second range is less than said first range;and

storing means for storing said transformed kernel.

According to a still further aspect of the present invention there isprovided a computer readable medium for storing a program for anapparatus which processes data, said processing comprising a method oftransforming a kernel for image processing, said program comprising:

code for accessing a kernel for image processing;

code for transforming said kernel from a first range to a second range,wherein said second range is less than said first range; and

code for storing said transformed kernel.

According to a still further aspect of the present invention there isprovided a method of converting a first set of discrete data samplevalues of an image having a first sample rate to a second set ofdiscrete data sample values of said image having a second sample rate,the method comprising the following steps:

(i) accessing said first set of data sample values; and

(ii) performing the following operations for each data value of saidsecond set;

(a) calculating kernel values for each of said discrete sample values ofsaid first set, according to a first kernel, wherein said first kernelis constructed from a scaled version of a second kernel; and

(b) convolving said kernel values with said discrete sample values ofsaid first data set to provide a current data value of said second set.

According to a still further aspect of the present invention there isprovided an apparatus for converting a first set of discrete data samplevalues of an image having a first sample rate to a second set ofdiscrete data sample values of said image having a second sample rate,the apparatus comprising:

access means for accessing said first set of data sample values; and

processing means for performing the following operations for each datavalue of said second set;

(a) calculating kernel values for each of said discrete sample values ofsaid first set, according to a first kernel, wherein said first kernelis constructed from a scaled version of a second kernel; and

(b) convolving said kernel values with said discrete sample values ofsaid first data set to provide a current data value of said second set.

According to a still further aspect of the present invention there isprovided a computer readable medium for storing a program for anapparatus which processes data, said processing comprising a method ofconverting a first set of discrete data sample values of an image havinga first sample rate to a second set of discrete data sample values ofsaid image having a second sample rate, said program comprising:

code for accessing said first set of data sample values; and

code for performing the following operations for each data value of saidsecond set;

(a) calculating kernel values for each of said discrete sample values ofsaid first set, according to a first kernel, wherein said first kernelis constructed from a scaled version of a second kernel; and

(b) convolving said kernel values with said discrete sample values ofsaid first data set to provide a current data value of said second set.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention are described with reference to thedrawings, in which:

FIG. 1 shows a coefficient value versus pixel plot for a nearestneighbour kernel;

FIG. 2 shows a coefficient value versus pixel plot for a linear kernel;

FIG. 3 shows a coefficient value versus pixel plot for a quadratickernel;

FIG. 4 shows a coefficient value versus pixel plot for a cubic kernel;

FIG. 5 shows a step response for a nearest neighbour kernel;

FIG. 6 shows a step response for a linear kernel;

FIG. 7 shows a step response for a quadratic kernel;

FIG. 8 shows a step response for a cubic kernel;

FIG. 9 is a flow diagram of a method of interpolating digital data inaccordance with a preferred embodiment of the present invention;

FIG. 10 shows a linear transform, of the cubic interpolation kernel ofFIG. 4, from the range s=[0,1] to the range s′=[d, 1−d];

FIG. 11 shows a linear transform of a modified cubic kernel with variousvalues of d in accordance with the present invention;

FIG. 12 shows a general purpose computer upon which the method of thepreferred embodiment can be practised;

FIG. 13 shows the step response of a modified cubic kernel with variousvalues of d in accordance with the present invention; and

FIG. 14 shows a graph of the dead-zone parameter d, for varying valuesof temporal difference, in accordance with the preferred embodiment ofthe present invention.

DETAILED DESCRIPTION

Where reference is made in any one or more of the drawings to stepsand/or features, which have the same reference numerals, those stepsand/or features are for the purposes of the description the same, unlessthe contrary appears.

FIG. 9 is a flow diagram of a method of resampling image data inaccordance with a preferred embodiment of the invention. The preferredmethod maintains the appearance of sharp lines and text at arbitraryre-sampling ratios. The method commences at step 102 where any necessaryprocesses and parameters are initialised, such as counters. At the nextstep 104, sample values in the form f(kΔt) (k=−n . . . −2, −1, 0, 1, 2,. . . n), where Δt is the (constant) sampling rate, are retrieved forprocessing. The process continues at decision block 106, where a checkis carried out to find out if the final sample value, at t=T, has beencalculated. If the final sample value has been calculated then theprocess will end. Otherwise, at the next step 108, kernel valuesh(T−kΔt) are calculated in accordance with the kernel of the preferredembodiment, h(s), given by: $\begin{matrix}{{h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.} & (3)\end{matrix}$

otherwise where s=t/Δt is a normalised coordinate that has integervalues at the sample points. In the preferred embodiment, the kernelparameter d has a default value of 0.2. Alternatively, the user canselect a desired value for d. Varying d allows the user to trade offedge sharpness with edge translation effects. In the next step 110, thesample values, f(kΔt), are convolved with the kernel values h(T−kΔt),according to the finite convolution sum:

g(T)=Σf(kΔt)h(T−kΔt),

at t=T. The process continues at step 112 where the result of theconvolution sum, g(T), is output. The process will continue until thefinal sample value has been calculated.

The following paragraphs provide a more detailed explanation of thekernel according to the preferred embodiment.

Interpolation with a continuous kernel effectively gives a continuousfunction g(t), that is an approximation of f(t), and is given by thefinite convolution sum:

g(t)=Σf(kΔt)h(t−kΔt).

The kernels are, by definition, of finite support and symmetrical abouts=0. The kernel of the preferred embodiment is a modified cubic kerneland can be shown to be an interpolating kernel, where the interpolatedfunction passes through the sample points of the original function, ash(0)=1, while h(kΔt)=0 for k≠0 which ensure that g(kΔt)=f(kΔt). FromEquation (4) any sample of g(t) can be calculated using the continuouskernel, h(t), and a finite number of original sample points, f(kΔt),adjacent to the sample being interpolated. For example, the conventionalcubic kernel, defined by equation (1), requires the four adjacent samplepoints while the modified cubic kernel, defined by Equation (3),requires only two.

The modified cubic kernel is what is known as a separable kernel. Thismeans that the kernel can be applied to the image data in one of twoequivalent ways:

(i) applying the convolution kernel to the rows of the image and thenusing these interpolated values to interpolate along the columns of theimage (or vice-versa). For example, the conventional cubic kernel hasfour coefficients and requires the four nearest samples for both therows and the columns. The modified cubic however, has only twocoefficients and requires only the two nearest samples. This techniquehowever, has the disadvantage of requiring intermediate storage ofinterpolated values, which can make it not suitable for a hardwareimplementation for example.

(ii) A 2-dimensional kernel can be generated and then convolved directlywith the image data. The 2-dimensional kernel is generated using matrixmultiplication of the coefficient values calculated for the rows and thecolumns separately. For example, the conventional cubic kernel has 16(4×4) coefficients while the modified cubic has only four, (2×2). These2-dimensional blocks of coefficients are then convolved directly withthe same size block of nearest neighbour image samples around the pixelbeing interpolated. This method has the advantage of not requiring anyintermediate storage, but does require more multiplication operations.

The preferred method of generating the modified cubic interpolationkernel is to linearly transform the cubic interpolation kernel definedin Equation (2) from the range s=[0,1] to the range s′=[d, 1−d], asshown in FIG. 10. This involves making the substitution s′=(s−d)/1−2d)so that the point s=d becomes s′=0 and the s=(1−d) becomes s′=1. In thisway, the original cubic is effectively compressed into the reduced range[d, 1−d]. This effectively creates a symmetrical “dead-zone” at both thelower and upper ends of the kernel as illustrated in FIG. 10. When d=0,the function reduces to a conventional cubic kernel (with b=0, c=0) andwhen d=0.5, the NN kernel results. As explained above, varying d allowsthe user to trade off edge sharpness with edge translation effects andgain the benefits of both the NN and cubic convolution whilst minimisingtheir disadvantages. The modified cubic kernel is shown with variousvalues of d in FIG. 11. In the preferred embodiment a constant value ofd=0.2 is used, however the scope of the preferred embodiment coversmethods that vary d depending on re-sampling ratio or edge proximity.FIG. 13 demonstrates the effect of varying d on the step response of themodified cubic kernel. For example, increasing d increases the slope ofthe step response and therefore increases the perceived edge sharpness.

The method is also particularly advantageous for interpolating imagesequences, for example when changing the frame (or field) rate of avideo sequence. This is because it only requires the two nearest framesto interpolate any intermediate frame, which minimises frame storage. Inaddition, the separable nature of the kernel means that the kernel canbe applied along the rows, columns, and along the temporal domain inturn or alternatively a 3-dimensional kernel can be constructed andconvolved the 3-dimensional video data. Various combinations of theabove two schemes, which trade off memory storage and computationalcomplexity, are also possible.

In another embodiment of the present invention the modified cubic kernelis adapted according to the temporal differences in a sequence ofimages, or video. At each pixel in the image sequence, I(x,y,t), thetemporal difference δ with the previous frame is calculated as follows:

δ=|I(x,y,t)−I(x,y,t−1)|  (5)

This is then used to adapt the dead-zone parameter, d, according to$\begin{matrix}{{d = {\min \left( {0.5,{\max \left( {0,{\frac{\left( {\delta - v} \right)}{m} + w}} \right)}} \right)}},} & \text{(6)}\end{matrix}$

where m is the slope parameter, where m>0, and (v,w) are the coordinatesof a point the function is constrained to pass through. Preferably,(v,w) are set to be (128, 0.25), and a slope parameter of m=320 is used,as illustrated in FIG. 14. In this way, when the temporal differencebetween pixels is small the conventional cubic kernel is used, whilstwhen temporal differences are large the nearest neighbor kernel is(effectively) used. The method then adapts the kernel shape with agradual transition between the cubic and NN kernels at moderate temporaldifferences. This adaptation reduces the effects of motion blur in videosequences that contain a large amount of motion.

The preferred embodiment is based on a modification of the cubic kerneldefined in Equation (2). It is realised that the technique disclosed canbe applied to any continuous kernel, such as the linear, quadratic, orweighted sinc kernels. However, the linear, quadratic, and sinc kernelshave fewer degrees of freedom than the cubic kernel and this results ina discontinuity, ie. a non-smooth transition, where the dead-zone joinsthe (transformed) original kernel. This discontinuity will thereforecause a discontinuity in the interpolated image data, which can not bedesirable. However, the modified linear and quadratic kernels will havean advantage in terms of a simplified implementation and reducedcomputational complexity. The modified linear kernel, h(s), is given by:$\begin{matrix}{{h(s)}\left\{ \quad \begin{matrix}1 & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > {1 - d}} \\{{1 - \frac{{s - d}}{1 - {2d}}},} & {elsewhere}\end{matrix}\quad \right.} & \text{(7)}\end{matrix}$

where s=t/Δt is a normalised coordinate that has integer values at thesample points.

The method can also be applied to colour images and video sequences. Forcolour images the kernel can be applied to each of the colour planesindependently, such as say RGB, YUV, YCbCr etc. If the modified cubickernel is used adaptively on sharp image edges only, the kernel can beapplied to all three colour planes when an edge is detected in any oneof the colour planes, i.e., edge strength is calculated as the maximumof the edge strength in the three colour channels.

The kernel of the preferred embodiment can be used in conjunction with aplurality of other continuous kernels each optimised for different imagefeatures, such as smooth areas, textures, or sharp edged. As alreadydescribed the modified cubic kernel disclosed is particularlyadvantageous when interpolating sharp edges. Therefore, a scheme thatuses the conventional cubic (b=0, c=0.5) in smooth image areas, eg. whenthe image gradient is below a predetermined threshold, and the modifiedcubic kernel disclosed in the current invention would offer goodperformance.

Preferred Embodiment of Apparatus(s)

The preferred method is preferably practiced using a conventionalgeneral-purpose computer system, such as the system 1200 shown in FIG.12, wherein the process of FIGS. 9 to 11 can be implemented as softwareexecuting on the computer. In particular, the steps of the method areeffected by instructions in the software that are carried out by thecomputer. The software can be divided into two separate parts; one partfor carrying out the method of the preferred embodiment; and anotherpart to manage the user interface between the latter and the user. Thesoftware can be stored in a computer readable medium, including thestorage devices described below, for example. The software is loadedinto the computer from the computer readable medium, and then executedby the computer. A computer readable medium having such software orcomputer program recorded on it is a computer program product. The useof the computer program product in the computer preferably effects anadvantageous apparatus for orientating a character stroke orn-dimensional finite space curves in accordance with the embodiments ofthe invention.

The computer system 1200 has a computer module 1202, a video display1216, and input devices 1218, 1220. In addition, the computer system1200 can have any of a number of other output devices including lineprinters, laser printers, plotters, and other reproduction devicesconnected to the computer module 1202. The computer system 1200 can beconnected to one or more other computers via a communication interface1208 c using an appropriate communication channel 1230 such as a modemcommunications path, a computer network, or the like. The computernetwork can include a local area network (LAN), a wide area network(WAN), an Intranet, and/or the Internet

The computer module 1202 has a central processing unit(s) (simplyreferred to as a processor hereinafter) 1204, a memory 1206 which caninclude random access memory (RAM) and read-only memory (ROM),input/output (IO) interfaces 1208, a video interface 1210, and one ormore storage devices generally represented by a block 1212 in FIG. 12.The storage device(s) 1212 can include of one or more of the following:a floppy disc, a hard disc drive, a magneto-optical disc drive, CD-ROM,magnetic tape or any other of a number of non-volatile storage deviceswell known to those skilled in the art. Each of the components 1204 to1212 is typically connected to one or more of the other devices via abus 1214 that in turn has data, address, and control buses.

The video interface 1210 is connected to the video display 1216 andprovides video signals from the computer 1202 for display on the videodisplay 1216. User input to operate the computer 1202 can be provided byone or more input devices 1208. For example, an operator can use thekeyboard 1218 and/or a pointing device such as the mouse 1220 to provideinput to the computer 1202.

The system 1200 is simply provided for illustrative purposes and otherconfigurations can be employed without departing from the scope andspirit of the invention. Exemplary computers on which the embodiment canbe practiced include the IBM-PC/ATs or compatibles, one of theMacintosh™ family of PCs, Sun Sparcstation™, arrangements evolvedtherefrom. The foregoing are merely exemplary of the types of computerswith which the embodiments of the invention can be practiced. Typically,the processes of the embodiments, described hereinafter, are resident assoftware or a program recorded on a hard disk drive (generally depictedas block 1212 in FIG. 12) as the computer readable medium, and read andcontrolled using the processor 1204. Intermediate storage of the programand pixel data and any data fetched from the network can be accomplishedusing the semiconductor memory 1206, possibly in concert with the harddisk drive 1212.

In some instances, the program can be supplied to the user encoded on aCD-ROM or a floppy disk (both generally depicted by block 1212), oralternatively could be read by the user from the network via a modemdevice connected to the computer, for example. Still further, thesoftware can also be loaded into the computer system 1200 from othercomputer readable medium including magnetic tape, a ROM or integratedcircuit, a magneto-optical disk, a radio or infra-red transmissionchannel between the computer and another device, a computer readablecard such as a PCMCIA card, and the Internet and Intranets includingemail transmissions and information recorded on websites and the like.The foregoing are merely exemplary of relevant computer readablemediums. Other computer readable mediums can be practiced withoutdeparting from the scope and spirit of the invention.

The preferred method can alternatively be implemented in dedicatedhardware such as one or more integrated circuits performing thefunctions or sub functions of the steps of the method. Such dedicatedhardware can include graphic processors, digital signal processors, orone or more microprocessors and associated memories.

The foregoing only describes one embodiment of the present invention,however, modifications and/or changes can be made thereto by a personskilled in the art without departing from the scope and spirit of theinvention.

What is claimed is:
 1. A method of providing a representation of imagedata, the method comprising the following steps: (i) accessing aplurality of discrete sample values of said image data; (ii) calculatingkernel values for each of said discrete sample values using a modifiedcontinuous kernel, said modified continuous kernel being constructed byshifting and scaling a continuous kernel, said continuous kernel beingscaled from a first range to a second range, said second range beingless than said first range, wherein said modified continuous kernel hasa substantially symmetrical dead-zone at both an upper end and a lowerend of said modified continuous kernel; and (iii) convolving said kernelvalues with said discrete sample values to provide a representation ofsaid image data.
 2. The method of according to claim 1, wherein saidcontinuous kernel is of the form: ${h(s)} = \left\{ \begin{matrix}{{{2{s}^{3}} - {3{s}^{2}} + 1},} & {{s} \leq 1} \\{0,} & {Otherwise}\end{matrix} \right.$

and wherein s=t/Δt.
 3. The method according to claim 1, wherein saidmodified continuous kernel is of the form:${h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.$

and wherein s=tΔt and 0≦d<0.5.
 4. The method according to claim 2,wherein said first range is s=[0,1] and said second range is s=[d, 1−d].5. The method according to claim 1, wherein said image data is a portionof a sequence of image data signals.
 6. The method according to claim 5,wherein a temporal difference between a previous image data signal and acurrent image data signal is calculated.
 7. The method according toclaim 3, wherein parameter d is varied according to a temporaldifference between a previous image data signal and a current image datasignal.
 8. The method according to claim 3, wherein parameter d isvaried according to a sampling ratio.
 9. The method according to claim3, wherein parameter d is varied according to edge proximity.
 10. Themethod according to claim 1, wherein said continuous kernel is a linearkernel.
 11. The method according to claim 1, wherein said continuouskernel is a quadratic kernel.
 12. The method according to claim 1,wherein said continuous kernel is a weighted sinc kernel.
 13. Anapparatus for providing a representation of image data, the apparatuscomprising: accessing means for accessing a plurality of discrete samplevalues of said image data; calculating means for calculating kernelvalues for each of said discrete sample values using a modifiedcontinuous kernel, said modified continuous kernel being constructed byshifting and scaling a continuous kernel, said continuous kernel beingscaled from a first range to a second range, said second range beingless than said first range, wherein said modified continuous kernel hasa substantially symmetrical dead-zone at both an upper end and a lowerend of said modified continuous kernel; and convolving means forconvolving said kernel values with said discrete sample values toprovide a representation of said image data.
 14. The apparatus accordingto claim 13, wherein said continuous kernel is of the form:${h(s)} = \left\{ \begin{matrix}{{{2{s}^{3}} - {3{s}^{2}} + 1},} & {{s} \leq 1} \\{0,} & {Otherwise}\end{matrix} \right.$

and wherein s=t/Δt.
 15. The apparatus according to claim 13, whereonsaid modified continuous kernel is of the form:${h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.$

and wherein s=t/Δt and 0≦d<0.5.
 16. The apparatus according to claim 14,wherein said first range is s=[0,1] and said second range is s=[d, 1−d].17. The apparatus according to claim 13, wherein said image data is aportion of a sequence of image data signals.
 18. The apparatus accordingto claim 17, wherein a temporal difference between a previous image datasignal and a current image data signal is calculated.
 19. The apparatusaccording to claim 15, wherein parameter d is varied according to atemporal difference between a previous image data current image datasignal.
 20. The apparatus according to claim 15, wherein parameter d isvaried according to a sampling ratio.
 21. The apparatus according toclaim 15, wherein parameter d is varied according to edge proximity. 22.The apparatus according to claim 13, wherein said continuous kernel is alinear kernel.
 23. The apparatus according to claim 13, wherein saidcontinuous kernel is a quadratic kernel.
 24. The apparatus according toclaim 13, wherein said continuous kernel is a weighted sinc kernel. 25.A computer readable medium storing a program for an apparatus whichprocesses data, said processing comprising a method of providing arepresentation of image data, said program comprising: code foraccessing a plurality of discrete sample values of said image data; codefor calculating kernel values for each of said discrete sample valuesusing a modified continuous kernel, said modified continuous kernelbeing constructed by shifting and scaling a continuous kernel, saidcontinuous kernel being scaled from a first range to a second range,said second range being less than said first range, wherein saidmodified continuous kernel has a substantially symmetrical dead-zone atboth an upper end and a lower end of said modified continuous kernel;and code for convolving said kernel values with said discrete samplevalues to provide a representation of said image data.
 26. The computerreadable medium of according to claim 25, wherein said continuous kernelis of the form: ${h(s)} = \left\{ \quad {\begin{matrix}{{{2{s}^{3}} - {3{s}^{2}} + 1},{{s} \leq 1}} \\{0,\quad {Otherwise}}\end{matrix}\quad,} \right.$

and wherein s=t/Δt.
 27. The computer readable medium according to claim25, wherein said modified continuous kernel is of the form:${h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.$

and wherein s=t/Δt and 0≦d<0.5.
 28. The computer readable mediumaccording to claim 26, wherein said first range is s=[0,1] and saidsecond range is s=[d, 1−d].
 29. The computer readable medium accordingto claim 25, wherein said image data is a portion of a sequence of imagedata signals.
 30. The computer readable medium according to any claim29, wherein a temporal difference between a previous image data signaland a current image data signal is calculated.
 31. The computer readablemedium according to claim 27, wherein parameter d is varied according toa temporal difference between a previous image data signal and a currentimage data signal.
 32. The computer readable medium according to claim27, wherein parameter is varied according to a sampling ratio.
 33. Thecomputer readable medium according to claim 27, wherein parameter d isvaried according to edge proximity.
 34. The computer readable mediumaccording to claim 25, wherein said continuous kernel is a linearkernel.
 35. The computer readable medium according to claim 25, whereinsaid continuous kernel is a quadratic kernel.
 36. The computer readablemedium according to claim 25, wherein said continuous kernel is aweighted sinc kernel.
 37. A method of representing image data, themethod comprising the following steps: (i) accessing a plurality ofdiscrete sample values of said image data; (ii) calculating kernelvalues for each of said discrete sample values using a scaled kernel,wherein said scaled kernel is of the form:${h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.$

and wherein s=t/Δt and 0≦d<0.5; and (iii) convolving said kernel valueswith said discrete sample values to provide a representation of saidimage data.
 38. The method according to claim 37, wherein said scaledkernel is generated by transforming a kernel from a first range to asecond range, and wherein said second range is less than said firstrange.
 39. The method of according to claim 38, wherein said kernel isof the form: ${h(s)} = \left\{ \begin{matrix}{{{2{s}^{3}} - {3{s}^{2}} + 1},} & {{s} \leq 1} \\{0,} & {Otherwise}\end{matrix} \right.$

and wherein s=t/Δt.
 40. The method according to claim 38, wherein saidfirst range is s=[0,1] and said second range is s=[d, 1−d].
 41. Anapparatus for representing image data, the apparatus comprising:accessing means for accessing a plurality of discrete sample values ofsaid image data; calculating means for calculating kernel values foreach of said discrete sample values using a scaled kernel, wherein saidscaled kernel is of the form: ${h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.$

and wherein s=t/Δt and 0≦d<0.5; and convolving means for convolving saidkernel values with said discrete sample values to provide arepresentation of said image data.
 42. The apparatus according to claim41, wherein said scaled kernel is generated by transforming a kernelfrom a first range to a second range, and said second range is less thansaid first range.
 43. The apparatus according to claim 42, wherein saidkernel is of the form: ${h(s)} = \left\{ \begin{matrix}{{{2{s}^{3}} - {3{s}^{2}} + 1},} & {{s} \leq 1} \\{0,} & {Otherwise}\end{matrix} \right.$

and wherein s=t/Δt.
 44. The apparatus according to claim 42, whereinsaid first range is s=[0,1] and said second range is s=[d, 1−d].
 45. Acomputer readable medium storing a program for an apparatus whichprocesses data, said processing comprising a method of representingimage data, said program comprising: code for accessing a plurality ofdiscrete sample values of said image data; code for calculating kernelvalues for each of said discrete sample values using a scaled kernel,wherein said scaled kernel is of the form:${h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{2 - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.$

and wherein s=t/Δt and 0≦d<0.5; and code for convolving said kernelvalues with said discrete sample values to provide a representation ofsaid image data.
 46. The computer readable medium according to claim 45,wherein said scaled kernel is generated by transforming a kernel from afirst range to a second range, and wherein said second range is lessthan said first range.
 47. The computer readable medium according toclaim 46, wherein said kernel is of the form:${h(s)} = \left\{ \begin{matrix}{{{2{s}^{3}} - {3{s}^{2}} + 1},} & {{s} \leq 1} \\{0,} & {Otherwise}\end{matrix} \right.$

and wherein s=t/Δt.
 48. The computer readable medium according to claim46, wherein said first range is s=[0,1] and said second range is s=[d,1−d].
 49. A method of transforming a kernel for image processing, themethod comprising the following steps: (i) accessing a continuous kernelfor image processing; (ii) shifting and scaling said continuous kernelfrom a first range to a second range to produce a modified continuouskernel, said second range being less than said first range, wherein saidmodified continuous kernel has a substantially symmetrical dead-zone atboth an upper end and a lower end of said modified continuous kernel;and (iii) storing said modified continuous kernel.
 50. The methodaccording to claim 49, wherein said continuous kernel is of the form:${h(s)} = \left\{ \begin{matrix}{{{2{s}^{3}} - {3{s}^{2}} + 1},} & {{s} \leq 1} \\{0,} & {Otherwise}\end{matrix} \right.$

and wherein s=t/Δt.
 51. The method according to claim 49, wherein saidmodified continuous kernel is of the form:${h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.$

and wherein s=t/Δt and 0≦d<0.5.
 52. The method according to claim 49,wherein said first range is s=[0,1] and said second range is s=[d, 1−d].53. The method according to claim 51, wherein parameter d is variedaccording to a temporal difference between a previous image data signaland current image data signal.
 54. The method according to claim 51,wherein varied according to a sampling ratio.
 55. The method accordingto claim 51, wherein parameter d is varied according to edge proximity.56. The method according to claim 49, wherein the continuous kernel is alinear kernel.
 57. The method according to claim 49, wherein thecontinuous kernel is a quadratic kernel.
 58. The method according toclaim 49, wherein the continuous kernel is a weighted sinc kernel. 59.An apparatus for transforming a kernel for image processing, saidapparatus comprising: accessing means for accessing a continuous kernelfor image processing; transforming means for shifting and scaling saidcontinuous kernel from a first range to a second range to produce amodified continuous kernel, said second range being less than said firstrange, wherein said modified continuous kernel has a substantiallysymmetrical dead-zone at both an upper end and a lower end of saidmodified continuous kernel; and storing means for storing said modifiedcontinuous kernel.
 60. The apparatus according to claim 59, wherein saidcontinuous kernel is of the form: ${h(s)} = \left\{ \begin{matrix}{{{2{s}^{3}} - {3{s}^{2}} + 1},} & {{s} \leq 1} \\{0,} & {Otherwise}\end{matrix} \right.$

and wherein s=t/Δt.
 61. The apparatus according to claim 59, whereinsaid modified kernel is of the form: ${h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{2 - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.$

and wherein s=t Δt and 0≦d<0.5.
 62. The apparatus according to claim 59,wherein said first range is s=[0,1] and said second range is s=[d, 1−d].63. The apparatus according to claim 61, wherein parameter d is variedaccording to a temporal difference between a previous image data signaland a current image data signal.
 64. The apparatus according to claim61, wherein parameter d is varied according to a sampling ratio.
 65. Theapparatus according to claim 61, wherein parameter d is varied accordingto edge proximity.
 66. The apparatus according to claim 59, wherein thecontinuous kernel is a linear kernel.
 67. The apparatus according toclaim 59, wherein the continuous kernel is a quadratic kernel.
 68. Theapparatus according to claim 59, wherein the continuous kernel is aweighted sinc kernel.
 69. A computer readable medium storing a programfor an apparatus which processes data, said processing comprising amethod of transforming a kernel for image processing, said programcomprising: code for accessing a continuous kernel for image processing;code for shifting and scaling said continuous kernel from a first rangeto a second range to produce a modified continuous kernel, said secondrange being less than said first range, wherein said modified continuouskernel has a substantially symmetrical dead-zone at both an upper endand a lower end of said modified continuous kernel; and code for storingsaid modified continuous kernel.
 70. The computer readable mediumaccording to claim 69, wherein said continuous kernel is of the form:${h(s)} = \left\{ \begin{matrix}{{{2{s}^{3}} - {3{s}^{2}} + 1},} & {{s} \leq 1} \\{0,} & {Otherwise}\end{matrix} \right.$

and wherein s=t/Δt.
 71. The computer readable medium according to claim69, wherein said modified continuous kernel is of the form:${h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.$

and wherein s=t/Δt and 0≦d<0.5.
 72. The computer readable mediumaccording to claim 69, wherein said first range is s=[0,1] and saidsecond range is s=[d, 1−d].
 73. The computer readable medium accordingto claim 71, wherein parameter d is varied according to a temporaldifference between a previous image data signal and a current image datasignal.
 74. The computer readable medium according to claim 71, hereinparameter d is varied according to a sampling ratio.
 75. The computerreadable medium according to claim 71, wherein parameter d is variedaccording to edge proximity.
 76. The computer readable medium accordingto claim 69, wherein the continuous kernel is a linear kernel.
 77. Thecomputer readable medium according to claim 69, wherein the continuouskernel is a quadratic kernel.
 78. The computer readable medium accordingto claim 69, wherein the continuous kernel is a weighted sinc kernel.79. A method of converting a first set of discrete data sample values ofan image having a first sample rate to a second set of discrete datasample values of said image having a second sample rate, the methodcomprising the following steps: (i) accessing said first set of datasample values; and (ii) performing the following operations for eachdata value of said second set; (a) calculating kernel values for each ofsaid discrete sample values of said first set, according to a modifiedcontinuous kernel, wherein said modified continuous kernel isconstructed from a shifted and scaled version of a second kernel,wherein said modified continuous kernel has a substantially symmetricaldead-zone at both an upper end and a lower end of said modifiedcontinuous kernel; and (b) convolving said kernel values with saiddiscrete sample values of said first data set to provide a current datavalue of said second set.
 80. The method of according to claim 79,wherein said second kernel is of the form:${h(s)} = \left\{ \quad {\begin{matrix}{{{2{s}^{3}} - {3{s}^{2}} + 1},{{s} \leq 1}} \\{0,\quad {Otherwise}}\end{matrix}\quad,} \right.$

and wherein s=t/Δt.
 81. The method according to claim 79, wherein saidmodified continuous kernel is of the form:${h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.$

and wherein s=t/Δ≦d<0.5.
 82. The method according to claim 79, whereinsaid image is sampled to generate said first set of discrete data samplevalues.
 83. The method according to claim 81, wherein parameter d isvaried according to a temporal difference between a previous image datasignal and a current image data signal.
 84. The method according toclaim 81, wherein parameter d is varied according to a sampling ratio.85. The method according to claim 81, wherein parameter d is variedaccording to edge proximity.
 86. The method according to claim 79,wherein the second kernel is a linear kernel.
 87. The method accordingto claim 79, wherein the second kernel is a quadratic kernel.
 88. Themethod according to claim 79, wherein the second kernel is a weightedsinc kernel.
 89. An apparatus for converting a first set of discretedata sample values of an image having a first sample rate to a secondset of discrete data sample values of said image having a second samplerate, the apparatus comprising: access means for accessing said firstset of data sample values; and processing means for performing thefollowing operations for each data value of said second set; (a)calculating kernel values for each of said discrete sample value of saidfirst set, according to a modified continuous kernel, wherein saidmodified continuous kernel is constructed from a shifted and scaledversion of a second kernel, wherein said modified continuous kernel hasa substantially symmetrical dead-zone at both an upper end and a lowerend of said modified continuous kernel; and (b) convolving said kernelvalues with said discrete sample values of said first data set toprovide a current data value of said second set.
 90. The apparatusaccording to claim 89, wherein said second kernel is of the form:${h(s)} = \left\{ \quad {\begin{matrix}{{{2{s}^{3}} - {3{s}^{2}} + 1},{{s} \leq 1}} \\{0,\quad {Otherwise}}\end{matrix}\quad,} \right.$

and wherein s=t/Δt.
 91. The apparatus according to claim 89, whereinsaid modified continuous kernel is of the form:${h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.$

and wherein s=t/Δt and 0≦d<0.5.
 92. The apparatus according to claim 89,wherein said image is sampled to generate said first set of discretedata sample values.
 93. The apparatus according to claim 91, whereinparameter d is varied according to a temporal difference between aprevious image data signal and a current image data signal.
 94. Theapparatus according to claim 91, wherein parameter d is varied accordingto a sampling ratio.
 95. The apparatus according to claim 91, whereinparameter d is varied according to edge proximity.
 96. The apparatusaccording to claim 89, wherein the second kernel is a linear kernel. 97.The apparatus according to claim 89, wherein the second kernel is aquadratic kernel.
 98. The apparatus according to claim 89, wherein thesecond kernel a weighted sinc kernel.
 99. A computer readable mediumstoring a program for an apparatus which processes data, said processingcomprising a method of converting a first set of discrete data samplevalues of an image having a first sample rate to a second set ofdiscrete data sample values of said image having a second sample rate,said program comprising: code for accessing said first set of datasample values; and code for performing the following operations for eachdata value of said second set; (a) calculating kernel values for each ofsaid discrete sample values of said first set, according to a modifiedcontinuous kernel, wherein said modified continuous kernel isconstructed from a shifted and scaled version of a second kernel,wherein said modified continuous kernel has a substantially symmetricaldead-zone at both an upper end and a lower end of said modifiedcontinuous kernel; and (b) convolving said kernel values with saiddiscrete sample values of said first data set to provide a current datavalue of said second set.
 100. The computer readable medium of accordingto claim 99, wherein said second kernel is of the form:${h(s)} = \left\{ \quad {\begin{matrix}{{{2{s}^{3}} - {3{s}^{2}} + 1},{{s} \leq 1}} \\{0,\quad {Otherwise}}\end{matrix}\quad,} \right.$

and wherein s=t/Δt.
 101. The computer readable medium according to claim99, wherein said modified continuous kernel is of the form:${h(s)} = \left\{ \begin{matrix}{1,} & {{- d} < s \leq d} \\{0,} & {{- \left( {1 - d} \right)} \geq s > \left( {1 - d} \right)} \\{{{2{\frac{s - d}{1 - {2d}}}^{3}} - {3{\frac{s - d}{1 - {2d}}}^{2}} + 1},} & {otherwise}\end{matrix} \right.$

and wherein s=t/Δt and 0≦d<0.5.
 102. The computer readable mediumaccording to claim 99, wherein said image is sampled to generate saidfirst set of discrete data sample values.
 103. The computer readablemedium according to claim 101, wherein parameter d is varied accordingto a temporal difference between a previous image data signal and acurrent image data signal.
 104. The computer readable medium accordingto claim 101, wherein parameter d is varied according to a samplingratio.
 105. The computer readable medium according to claim 101, whereinparameter d is varied according to edge proximity.
 106. The computerreadable medium according to claim 99, wherein the second kernel is alinear kernel.
 107. The computer readable medium according to claim 99,wherein the second kernel a quadratic kernel.
 108. The computer readablemedium according to claim 99, wherein the second kernel is a weightedsinc kernel.